Relaxation of one-dimensional nonlocal supremal functionals in the Sobolev setting

2023/07/25 by Torricelli, Andrea, Zappale, Elvira
#26B25 (secondary) #47J22 #49J45 (primary) #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.13597

Abstract

We provide necessary and sufficient conditions on the density W:\mathbb R^d×\mathbb R ^d→\mathbb R in order to ensure the sequential weak* lower semicontinuity of the functional J: W^1,∞(I;\mathbb R^d)→ \mathbb R, defined as J(u):=ess sup_{I× I}W(u'(x), u'(y)), when I is an open and bounded interval of \mathbb R. We also show that, when d=1, the lower semicontinuous envelope of I in general can be obtained by replacing W by its separately level convex envelope.

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